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11through the resonance regions during the descent in the rarefied atmosphere of Mars of a rigid body with con-
siderable geometric and aerodynamic asymmetries. The veracity of the obtained estimate is confirmed by the
Monte-Carlo method.
References
1. Neishtadt A. Averaging, capture into resonances, and chaos in nonlinear system, in Chaos. New York: Amer. Inst.
Phys., 1990. P. 261�273.
2. Kurkina E.V., Lyubimov V.V. Estimation of the probability of capture into resonance and parametric analysis in the
descent of an asymmetric spacecraft in an atmosphere // J. of Applied and Industrial Mathematics. 2018. V. 12, N. 3.
P. 492-500.
Stochastic modelling of financial securities with a systemic risk component
R. N. Makarov
Wilfrid Laurier University, Waterloo, ON, Canada
Email: rmakarov@wlu.ca
DOI 10.24412/cl-35065-2021-1-00-84
We propose and study a new jump-diffusion model for pricing multiple assets, where systemic-risk securi-
ty is combined with several conditionally independent base assets. This approach allows for analyzing and
modelling a portfolio that integrates high-activity security, such as an Exchange Trading Fund (ETF) tracking a
major market index (e. g., S&P500 or TSX) with several low-activity assets. The latter may have missing and
asynchronous pricing data when the assets are not traded frequently on financial markets. The proposed
framework allows for constructing several models, including the following: (a) a diffusion-type model without
jumps where all asset price processes are Geometric Brownian Motions; (b) a jump-diffusion model with only
common jumps [1]; (c) a jump-diffusion model with both common and asset-specific jumps [2]. We discuss the
properties of the proposed model, the estimation of its parameters using the Maximum Likelihood Estimation
method, and the pricing of European-style basket options. The NSERC Discovery Grant program supported this
work.
References
1. Chen Y., Makarov R. N. Modelling asynchronous assets with jump-diffusion processes. In International Conference
on Applied Mathematics, Modeling and Computational Science, pp. 477�487. Springer, 2017.
2. Xu R., Makarov R.N. High-Frequency Statistical Modelling for Jump-Diffusion Multi-Asset Price Processes with a
Systemic Component, Springer, 2019 (accepted).
Monte Carlo simulation of halos in crystal clouds
Q. Mu1, E. G. Kablukova2, B. A. Kargin1,2, S. M. Prigarin1,2
1Novosibirsk State University
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: mutsyuev@gmail.com
DOI 10.24412/cl-35065-2021-1-00-86
In this paper, we try to answer the question: how multiple scattering, sun elevation, shape and orientation
of ice crystals in the cirrus clouds affect a halo pattern. To study the radiation transfer in optically anisotropic
clouds we have developed the software based on Monte Carlo method [1] and ray tracing. In addition to halos,
this software enables one to simulate �anti-halos�, which above the cloud layer can be seen by observers [2].
We present the visualization of halos and anti-halos generated by the cirrus clouds for different shapes and
orientation of ice crystals.
The study was carried out under the CSC (China Scholarship Council) and the State Contract with ICMMG SB RAS
(0251-2021-0002).
References
1. Marchuk G. A., Mikhailov G. I., Nazaraliev M. A., Darbinian R. A., Kargin B. A., Elepov B. S. Monte Carlo methods in
atmospheric optics. Springer-Verlag, Berlin, Heidelberg, New York, 1980.
2. Tape W. Atmospheric Halos. M.: American Geophysical Union, 1994.
Study of asymptotics of particle transfer process with multiplication in a random medium
S. A. Rozhenko
The Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: sergroj@mail.ru
DOI 10.24412/cl-35065-2021-1-00-87
Simulation is carried out using weight modeling and double randomization in order to estimate the aver-
age particle flow from a random medium in which particle multiplication occurs.
The main goal of this work is to study the possibility of superexponential asymptotics being realized for a
standard model of an isotropic random field of density of the medium.
At the same time, for small correlation radii, a radical reduction in the complexity of calculations could be
achieved by replacing double randomization with randomized modeling of trajectories taking into account the
value of the correlation length.
This work was carried out under state contract with ICMMG SB RAS number 0251-2021-0002.
A vector Monte Carlo algorithm for large systems of linear equations
K. K. Sabelfeld1,2, A. E. Kireeva1
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: karl@osmf.sscc.ru, kireeva@ssd.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-89
A Monte Carlo randomization algorithm for solving large systems of linear algebraic equations is present-
ed. This algorithm combines the randomized stochastic matrix based algorithms proposed in [1], and an itera-
tive method of solving integral equations suggested in [2] which has no spectral restrictions on its convergens
in contrast to the conventional Neumann series based method. We develop a special transform of the original
non-negative matrix to a column stochastic matrix which is then conveniently used for calculation of matrix
iterations. The algorithm of randomized calculation of matrix iterations proposed in [1] operates by sampling
random rows and columns instead of matrix-matrix and matrix-vector multiplications. To solve a system of
linear algebraic equations with a matrix whose eigenvalues are greater than 1, we apply the transformation
and the relevant iterative procedure given in [2]. We analyze the correctness, laboriousness and efficiency of
the method for various matrix sizes. As a byproduct, a vector random walk on grids and a modified random
walk on boundary algorithms for three-dimensional potential problems are constructed.
This work was supported by the Russian Science Foundation under grant � 19-11-00019, and the Russian Fund of
Fundamental Studies under Grant 20-51-18009 in the part of random walk process implementations.
